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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.5.27a
Chapter 6, Problem 6.5.27a

21–30. {Use of Tech} Arc length by calculator


a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 
y = cos 2x, for 0 ≤ x ≤ π

Verified step by step guidance
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Recall the formula for the arc length of a curve defined by y = f(x) from x = a to x = b: \[L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\]
Identify the function and interval: here, \[y = \cos 2x\] with \[0 \leq x \leq \pi\]
Find the derivative of y with respect to x: \[\frac{dy}{dx} = \frac{d}{dx}(\cos 2x) = -2 \sin 2x\]
Square the derivative: \[\left(\frac{dy}{dx}\right)^2 = (-2 \sin 2x)^2 = 4 \sin^2 2x\]
Write the integral for the arc length using the formula: \[L = \int_{0}^{\pi} \sqrt{1 + 4 \sin^2 2x} \, dx\] This integral represents the arc length of the curve on the given interval.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Arc Length Formula

The arc length of a curve y = f(x) from x = a to x = b is given by the integral ∫ from a to b of √(1 + (dy/dx)²) dx. This formula calculates the length of the curve by summing infinitesimal line segments along the curve.
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Arc Length of Parametric Curves

Derivative of the Function

To apply the arc length formula, you need the derivative dy/dx of the function y = cos(2x). Differentiating y with respect to x gives dy/dx = -2 sin(2x), which is essential for substituting into the arc length integral.
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Derivatives of Other Trig Functions

Simplifying the Integral Expression

After finding dy/dx, substitute it into the arc length integral and simplify the expression under the square root. Simplification may involve trigonometric identities to make the integral easier to evaluate, especially when using a calculator.
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Simplifying Trig Expressions
Related Practice
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